Spahn category theoretic aspects of the theory of group schemes

(Group) functors and affine (group) schemes

Let $k$ be a ring. Let $k.Ring$ denote the category of $k$-rings. Let $k.Fun$ denote the category of (contravariant) functors $X:k:Ring\to Set$. Let $k.Aff$ denote the category of representable $k$-functors; we call this category the category of affine $k$-schemes and an object of this category we write as

$$Spec_k A:\begin{cases} k.Ring\to Set \\ R\mapsto hom(A,R) \end{cases}$$

We obtain in this way a functor

$$Spec_k:k.Ring\to k.Fun$$

This functor has a left adjoint

$$(O_k\dashv Spec_k):k.Ring\stackrel{Spec_k}{\to} k.Fun$$

assigning to a $k$-functor its ring of functions. This adjunction restricts to an adjoint equivalence

$$(O_k\dashv Spec_k):k.Ring\stackrel{Spec_k}{\to} k.Aff$$

and it restricts moreover to an adjoint equivalence

$$(O_k\dashv Spec_k):k.Bi Ring\stackrel{Spec_k}{\to} k.Aff.comm.Gr$$

between the categories of $k$-birings and the category of commutative affine $k$-group schemes. To see this be aware that a $k$-biring is a commutative ring object in ${k.Ring.comm}^{op}\simeq k.Aff.Sch$ (where the latter denotes the category of affine schemes).

(Group) schemes

A $k$-functor is called a $k$-scheme if it is a sheaf for the Zariski Grothendieck topology on $k.Ring^{op}$.

We will consider the moral of this op-ing below.

To give more details, recall that the closed sets of the Zariski topology on the spectrum $Spec A$ of a $k$-ring $A$ is defined by

$$V(I):=\{P\in Spec \, A|I\subseteq P\}$$

We can characterize the the elements of $V(a)$ also by

$$e_a(P)=0\, iff\, P\in V(a)$$

where

$$e_a:\begin{cases} Spec (A)\to Quot(A/P) \\ P\mapsto \frac{a\,mod\,P}{1} \end{cases}$$

where $Quot(A/P)$ denotes the quotient field (aka. field of fractions) of the integral domain $A/P$.

This construction generalizes to $k$-functors by defining an open subfunctor of a $k$-functor $X$ by

$$V(I):R\mapsto\{x\in X(R)| I\subseteq x\}$$

where $I\subseteq O(X)$. By the above alternative characterization, the assigned set consists precisely of those $x$ for which $f(x)=0$ for all $f\in I$.

Constant (group) scheme

$Sch_k$ is copowered (= tensored) over $Set$. We define the constant $k$-scheme on a set $E$ by

$$E_k:=E\otimes Sp_k k=\coprod_{e\in E}Sp_k k$$

For a scheme $X$ we compute $M_k(E_k,E)=Set(Sp_k k,X)^E=X(k)^E=Set(E,X(k))$ and see that there is an adjunction

$$((-)_k\dashv (-)(k)):Sch_k\to Set$$

If $E$ is a group $E_k$ is a group scheme.

We denote the category of $k$-schemes by $k.Sch$.

A $k$-group scheme is a group object in $k.Sch$. The category of $k$-group schemes we denote by $k.Grp$.

Formal (group) scheme

Let $k$ be a field. A finite $k$-ring is defined to be a $k$-algebra which is a finite dimensional $k$-vector space. The category of finite $k$-rings we denote by $fin.k.Ring$. A finite $k$-functor is defined to be a covariant functor $X:fin.k.Ring\to Set$. The category of finite $k$-functors we denote by $fin.k.Fun$. A finite $k$-scheme is defined to be $k$-scheme which is a finite $k$-functor. The category of finite $k$-schemes we denote by $fin.k.Sch$. Analogously we define the category of finite group schemes.

A formal group scheme is defined to be a codirected colimit of finite $k$-schemes.

Recall that we have a covariant embedding

$$Spec_k:k.Ring^{op}\to k.Fun$$

but we equivalently an embedding

$$Spec^*:k.co Ring\to k.Fun$$

where by $k.co Ring$ we denote the category of $k$-corings. A coring is a comonoid in the category of affine schemes (the latter is the opposite category of $k.Ring$). If we restrict to finite $k$-rings by linear algebra we have a bijection $A^*\otimes R=hom(A,k)\simeq hom(A,R)$ and can write

$$Spec_k C: R\mapsto\{u\in A^*\otimes R|\Delta_R u=u\otimes u, \epsilon_R u=1\}$$

where $C$ is a $k$-coring, $R$ a finite $k$-ring, $\Delta_R$ the skalar-extended comultiplication, $\epsilon_R$ the skalar-extended counit.

Examples of (group) schemes

examples of (group) schemes

Étale (group) scheme

(see also Grothendieck's Galois theory)

An étale group scheme over a field $k$ is defined to be a directed colimit

$$colim_{(k\hookrightarrow k^\prime)\in T\subseteq Sep} Spec\, k^\prime$$

where $T$ denotes some set of finite separable field extensions of $k$.

Cartier dual of a finite flat commutative group scheme

Let $G$ be a commutative $k$-group functor (in cases of interest this is a finite flat commutative group scheme). Then the Cartier dual $D(G)$ of $G$ is defined by

$$D(G)(R):=Gr_R(G\otimes_k R,\mu_R)$$

where $\mu_k$ denotes the group scheme assigning to a ring its multiplicative group $R^\times$ consisting of the invertible elements of $R$.

This definition deserves the name duality since we have

$$hom(G,D(H))=hom(H,D(G))=hom(G\times H,\mu_k)$$

p-torsion

Let $s:R\to S$ be a morphism of rings. Then we have an adjunction

$$(s^*\dashv s_*):S.Mod\stackrel{s_*}{\to} R.Mod$$

from the category of $S$-modules to that of $R$-modules where

$$s^*:A\mapsto A\otimes_s S$$

is called scalar extension and $s_*$ is called scalar restriction.

If $X$ denotes some scheme over a $k$-ring for $k$ being a field of characteristic $p$, we define its $p$-torsion component-wise by $X^{(p)}(R):=X(s_* R)$.

p-divisible groups

Witt rings and Dieudonné modules